LMFDB - Embedded newform 9450.2.a.eo.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9450,2,Mod(1,9450)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9450.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$75.4586299101$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{73}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 18 $ x^2 - x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1890)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.77200$ of defining polynomial
Character	$\chi$	$=$	9450.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} +3.77200 q^{11} -5.77200 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.77200 q^{17} +2.00000 q^{19} +3.77200 q^{22} -3.77200 q^{23} -5.77200 q^{26} -1.00000 q^{28} -3.00000 q^{29} -8.54400 q^{31} +1.00000 q^{32} +3.77200 q^{34} -8.77200 q^{37} +2.00000 q^{38} -6.77200 q^{41} +1.77200 q^{43} +3.77200 q^{44} -3.77200 q^{46} +4.54400 q^{47} +1.00000 q^{49} -5.77200 q^{52} -6.00000 q^{53} -1.00000 q^{56} -3.00000 q^{58} +6.77200 q^{59} +2.77200 q^{61} -8.54400 q^{62} +1.00000 q^{64} -9.54400 q^{67} +3.77200 q^{68} -6.77200 q^{71} -14.7720 q^{73} -8.77200 q^{74} +2.00000 q^{76} -3.77200 q^{77} -1.77200 q^{79} -6.77200 q^{82} +1.77200 q^{86} +3.77200 q^{88} +13.5440 q^{89} +5.77200 q^{91} -3.77200 q^{92} +4.54400 q^{94} +17.5440 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} - q^{11} - 3 q^{13} - 2 q^{14} + 2 q^{16} - q^{17} + 4 q^{19} - q^{22} + q^{23} - 3 q^{26} - 2 q^{28} - 6 q^{29} + 2 q^{32} - q^{34} - 9 q^{37} + 4 q^{38} - 5 q^{41} - 5 q^{43} - q^{44} + q^{46} - 8 q^{47} + 2 q^{49} - 3 q^{52} - 12 q^{53} - 2 q^{56} - 6 q^{58} + 5 q^{59} - 3 q^{61} + 2 q^{64} - 2 q^{67} - q^{68} - 5 q^{71} - 21 q^{73} - 9 q^{74} + 4 q^{76} + q^{77} + 5 q^{79} - 5 q^{82} - 5 q^{86} - q^{88} + 10 q^{89} + 3 q^{91} + q^{92} - 8 q^{94} + 18 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 - q^11 - 3 * q^13 - 2 * q^14 + 2 * q^16 - q^17 + 4 * q^19 - q^22 + q^23 - 3 * q^26 - 2 * q^28 - 6 * q^29 + 2 * q^32 - q^34 - 9 * q^37 + 4 * q^38 - 5 * q^41 - 5 * q^43 - q^44 + q^46 - 8 * q^47 + 2 * q^49 - 3 * q^52 - 12 * q^53 - 2 * q^56 - 6 * q^58 + 5 * q^59 - 3 * q^61 + 2 * q^64 - 2 * q^67 - q^68 - 5 * q^71 - 21 * q^73 - 9 * q^74 + 4 * q^76 + q^77 + 5 * q^79 - 5 * q^82 - 5 * q^86 - q^88 + 10 * q^89 + 3 * q^91 + q^92 - 8 * q^94 + 18 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	3.77200	1.13730	0.568651	−	0.822579i	$-0.307466\pi$
$11$	3.77200	1.13730	0.568651	+	0.822579i	$0.307466\pi$
$12$	0	0
$13$	−5.77200	−1.60087	−0.800433	−	0.599423i	$-0.795397\pi$
$13$	−5.77200	−1.60087	−0.800433	+	0.599423i	$0.795397\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	3.77200	0.914845	0.457422	−	0.889250i	$-0.348773\pi$
$17$	3.77200	0.914845	0.457422	+	0.889250i	$0.348773\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	3.77200	0.804194
$23$	−3.77200	−0.786517	−0.393258	−	0.919428i	$-0.628652\pi$
$23$	−3.77200	−0.786517	−0.393258	+	0.919428i	$0.628652\pi$
$24$	0	0
$25$	0	0
$26$	−5.77200	−1.13198
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−8.54400	−1.53455	−0.767274	−	0.641319i	$-0.778387\pi$
$31$	−8.54400	−1.53455	−0.767274	+	0.641319i	$0.778387\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.77200	0.646893
$35$	0	0
$36$	0	0
$37$	−8.77200	−1.44211	−0.721054	−	0.692879i	$-0.756342\pi$
$37$	−8.77200	−1.44211	−0.721054	+	0.692879i	$0.756342\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	0	0
$41$	−6.77200	−1.05761	−0.528805	−	0.848744i	$-0.677360\pi$
$41$	−6.77200	−1.05761	−0.528805	+	0.848744i	$0.677360\pi$
$42$	0	0
$43$	1.77200	0.270228	0.135114	−	0.990830i	$-0.456860\pi$
$43$	1.77200	0.270228	0.135114	+	0.990830i	$0.456860\pi$
$44$	3.77200	0.568651
$45$	0	0
$46$	−3.77200	−0.556151
$47$	4.54400	0.662811	0.331406	−	0.943488i	$-0.392477\pi$
$47$	4.54400	0.662811	0.331406	+	0.943488i	$0.392477\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	−5.77200	−0.800433
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−3.00000	−0.393919
$59$	6.77200	0.881640	0.440820	−	0.897596i	$-0.354688\pi$
$59$	6.77200	0.881640	0.440820	+	0.897596i	$0.354688\pi$
$60$	0	0
$61$	2.77200	0.354918	0.177459	−	0.984128i	$-0.443212\pi$
$61$	2.77200	0.354918	0.177459	+	0.984128i	$0.443212\pi$
$62$	−8.54400	−1.08509
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−9.54400	−1.16599	−0.582993	−	0.812477i	$-0.698118\pi$
$67$	−9.54400	−1.16599	−0.582993	+	0.812477i	$0.698118\pi$
$68$	3.77200	0.457422
$69$	0	0
$70$	0	0
$71$	−6.77200	−0.803689	−0.401844	−	0.915708i	$-0.631631\pi$
$71$	−6.77200	−0.803689	−0.401844	+	0.915708i	$0.631631\pi$
$72$	0	0
$73$	−14.7720	−1.72893	−0.864466	−	0.502691i	$-0.832343\pi$
$73$	−14.7720	−1.72893	−0.864466	+	0.502691i	$0.832343\pi$
$74$	−8.77200	−1.01972
$75$	0	0
$76$	2.00000	0.229416
$77$	−3.77200	−0.429860
$78$	0	0
$79$	−1.77200	−0.199366	−0.0996829	−	0.995019i	$-0.531783\pi$
$79$	−1.77200	−0.199366	−0.0996829	+	0.995019i	$0.531783\pi$
$80$	0	0
$81$	0	0
$82$	−6.77200	−0.747843
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	1.77200	0.191080
$87$	0	0
$88$	3.77200	0.402097
$89$	13.5440	1.43566	0.717831	−	0.696218i	$-0.245135\pi$
$89$	13.5440	1.43566	0.717831	+	0.696218i	$0.245135\pi$
$90$	0	0
$91$	5.77200	0.605070
$92$	−3.77200	−0.393258
$93$	0	0
$94$	4.54400	0.468678
$95$	0	0
$96$	0	0
$97$	17.5440	1.78132	0.890662	−	0.454666i	$-0.150241\pi$
$97$	17.5440	1.78132	0.890662	+	0.454666i	$0.150241\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	0	0
$101$	17.3160	1.72301	0.861503	−	0.507752i	$-0.169523\pi$
$101$	17.3160	1.72301	0.861503	+	0.507752i	$0.169523\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−5.77200	−0.565991
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−18.7720	−1.81476	−0.907379	−	0.420313i	$-0.861920\pi$
$107$	−18.7720	−1.81476	−0.907379	+	0.420313i	$0.861920\pi$
$108$	0	0
$109$	0.455996	0.0436765	0.0218383	−	0.999762i	$-0.493048\pi$
$109$	0.455996	0.0436765	0.0218383	+	0.999762i	$0.493048\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	16.5440	1.55633	0.778164	−	0.628061i	$-0.216151\pi$
$113$	16.5440	1.55633	0.778164	+	0.628061i	$0.216151\pi$
$114$	0	0
$115$	0	0
$116$	−3.00000	−0.278543
$117$	0	0
$118$	6.77200	0.623413
$119$	−3.77200	−0.345779
$120$	0	0
$121$	3.22800	0.293454
$122$	2.77200	0.250965
$123$	0	0
$124$	−8.54400	−0.767274
$125$	0	0
$126$	0	0
$127$	−7.22800	−0.641381	−0.320691	−	0.947184i	$-0.603915\pi$
$127$	−7.22800	−0.641381	−0.320691	+	0.947184i	$0.603915\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	−9.54400	−0.824476
$135$	0	0
$136$	3.77200	0.323446
$137$	−12.7720	−1.09119	−0.545593	−	0.838050i	$-0.683695\pi$
$137$	−12.7720	−1.09119	−0.545593	+	0.838050i	$0.683695\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	0	0
$142$	−6.77200	−0.568294
$143$	−21.7720	−1.82067
$144$	0	0
$145$	0	0
$146$	−14.7720	−1.22254
$147$	0	0
$148$	−8.77200	−0.721054
$149$	−11.3160	−0.927043	−0.463522	−	0.886086i	$-0.653414\pi$
$149$	−11.3160	−0.927043	−0.463522	+	0.886086i	$0.653414\pi$
$150$	0	0
$151$	1.31601	0.107095	0.0535475	−	0.998565i	$-0.482947\pi$
$151$	1.31601	0.107095	0.0535475	+	0.998565i	$0.482947\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	−3.77200	−0.303957
$155$	0	0
$156$	0	0
$157$	15.3160	1.22235	0.611175	−	0.791495i	$-0.290697\pi$
$157$	15.3160	1.22235	0.611175	+	0.791495i	$0.290697\pi$
$158$	−1.77200	−0.140973
$159$	0	0
$160$	0	0
$161$	3.77200	0.297275
$162$	0	0
$163$	−10.2280	−0.801119	−0.400559	−	0.916271i	$-0.631184\pi$
$163$	−10.2280	−0.801119	−0.400559	+	0.916271i	$0.631184\pi$
$164$	−6.77200	−0.528805
$165$	0	0
$166$	0	0
$167$	−5.22800	−0.404555	−0.202277	−	0.979328i	$-0.564834\pi$
$167$	−5.22800	−0.404555	−0.202277	+	0.979328i	$0.564834\pi$
$168$	0	0
$169$	20.3160	1.56277
$170$	0	0
$171$	0	0
$172$	1.77200	0.135114
$173$	0.772002	0.0586942	0.0293471	−	0.999569i	$-0.490657\pi$
$173$	0.772002	0.0586942	0.0293471	+	0.999569i	$0.490657\pi$
$174$	0	0
$175$	0	0
$176$	3.77200	0.284325
$177$	0	0
$178$	13.5440	1.01517
$179$	−1.54400	−0.115404	−0.0577021	−	0.998334i	$-0.518377\pi$
$179$	−1.54400	−0.115404	−0.0577021	+	0.998334i	$0.518377\pi$
$180$	0	0
$181$	4.31601	0.320806	0.160403	−	0.987052i	$-0.448721\pi$
$181$	4.31601	0.320806	0.160403	+	0.987052i	$0.448721\pi$
$182$	5.77200	0.427849
$183$	0	0
$184$	−3.77200	−0.278076
$185$	0	0
$186$	0	0
$187$	14.2280	1.04045
$188$	4.54400	0.331406
$189$	0	0
$190$	0	0
$191$	−15.0880	−1.09173	−0.545865	−	0.837873i	$-0.683799\pi$
$191$	−15.0880	−1.09173	−0.545865	+	0.837873i	$0.683799\pi$
$192$	0	0
$193$	−6.45600	−0.464713	−0.232356	−	0.972631i	$-0.574644\pi$
$193$	−6.45600	−0.464713	−0.232356	+	0.972631i	$0.574644\pi$
$194$	17.5440	1.25959
$195$	0	0
$196$	1.00000	0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	25.3160	1.79460	0.897302	−	0.441417i	$-0.145524\pi$
$199$	25.3160	1.79460	0.897302	+	0.441417i	$0.145524\pi$
$200$	0	0
$201$	0	0
$202$	17.3160	1.21835
$203$	3.00000	0.210559
$204$	0	0
$205$	0	0
$206$	−8.00000	−0.557386
$207$	0	0
$208$	−5.77200	−0.400216
$209$	7.54400	0.521830
$210$	0	0
$211$	10.3160	0.710183	0.355092	−	0.934832i	$-0.384450\pi$
$211$	10.3160	0.710183	0.355092	+	0.934832i	$0.384450\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−18.7720	−1.28323
$215$	0	0
$216$	0	0
$217$	8.54400	0.580005
$218$	0.455996	0.0308840
$219$	0	0
$220$	0	0
$221$	−21.7720	−1.46454
$222$	0	0
$223$	2.45600	0.164466	0.0822328	−	0.996613i	$-0.473795\pi$
$223$	2.45600	0.164466	0.0822328	+	0.996613i	$0.473795\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	16.5440	1.10049
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	0	0
$229$	−5.54400	−0.366358	−0.183179	−	0.983080i	$-0.558639\pi$
$229$	−5.54400	−0.366358	−0.183179	+	0.983080i	$0.558639\pi$
$230$	0	0
$231$	0	0
$232$	−3.00000	−0.196960
$233$	−24.7720	−1.62287	−0.811434	−	0.584444i	$-0.801313\pi$
$233$	−24.7720	−1.62287	−0.811434	+	0.584444i	$0.801313\pi$
$234$	0	0
$235$	0	0
$236$	6.77200	0.440820
$237$	0	0
$238$	−3.77200	−0.244503
$239$	21.8600	1.41401	0.707003	−	0.707210i	$-0.250047\pi$
$239$	21.8600	1.41401	0.707003	+	0.707210i	$0.250047\pi$
$240$	0	0
$241$	−19.7720	−1.27363	−0.636813	−	0.771018i	$-0.719748\pi$
$241$	−19.7720	−1.27363	−0.636813	+	0.771018i	$0.719748\pi$
$242$	3.22800	0.207504
$243$	0	0
$244$	2.77200	0.177459
$245$	0	0
$246$	0	0
$247$	−11.5440	−0.734527
$248$	−8.54400	−0.542545
$249$	0	0
$250$	0	0
$251$	−2.22800	−0.140630	−0.0703150	−	0.997525i	$-0.522400\pi$
$251$	−2.22800	−0.140630	−0.0703150	+	0.997525i	$0.522400\pi$
$252$	0	0
$253$	−14.2280	−0.894507
$254$	−7.22800	−0.453525
$255$	0	0
$256$	1.00000	0.0625000
$257$	−24.8600	−1.55072	−0.775362	−	0.631517i	$-0.782433\pi$
$257$	−24.8600	−1.55072	−0.775362	+	0.631517i	$0.782433\pi$
$258$	0	0
$259$	8.77200	0.545066
$260$	0	0
$261$	0	0
$262$	−3.00000	−0.185341
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	0	0
$266$	−2.00000	−0.122628
$267$	0	0
$268$	−9.54400	−0.582993
$269$	0.683994	0.0417039	0.0208519	−	0.999783i	$-0.493362\pi$
$269$	0.683994	0.0417039	0.0208519	+	0.999783i	$0.493362\pi$
$270$	0	0
$271$	−27.4040	−1.66468	−0.832338	−	0.554269i	$-0.812998\pi$
$271$	−27.4040	−1.66468	−0.832338	+	0.554269i	$0.812998\pi$
$272$	3.77200	0.228711
$273$	0	0
$274$	−12.7720	−0.771585
$275$	0	0
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−10.0000	−0.599760
$279$	0	0
$280$	0	0
$281$	−25.5440	−1.52383	−0.761914	−	0.647678i	$-0.775740\pi$
$281$	−25.5440	−1.52383	−0.761914	+	0.647678i	$0.775740\pi$
$282$	0	0
$283$	−4.31601	−0.256560	−0.128280	−	0.991738i	$-0.540946\pi$
$283$	−4.31601	−0.256560	−0.128280	+	0.991738i	$0.540946\pi$
$284$	−6.77200	−0.401844
$285$	0	0
$286$	−21.7720	−1.28741
$287$	6.77200	0.399739
$288$	0	0
$289$	−2.77200	−0.163059
$290$	0	0
$291$	0	0
$292$	−14.7720	−0.864466
$293$	3.68399	0.215221	0.107611	−	0.994193i	$-0.465680\pi$
$293$	3.68399	0.215221	0.107611	+	0.994193i	$0.465680\pi$
$294$	0	0
$295$	0	0
$296$	−8.77200	−0.509862
$297$	0	0
$298$	−11.3160	−0.655519
$299$	21.7720	1.25911
$300$	0	0
$301$	−1.77200	−0.102136
$302$	1.31601	0.0757276
$303$	0	0
$304$	2.00000	0.114708
$305$	0	0
$306$	0	0
$307$	33.3160	1.90144	0.950722	−	0.310043i	$-0.100344\pi$
$307$	33.3160	1.90144	0.950722	+	0.310043i	$0.100344\pi$
$308$	−3.77200	−0.214930
$309$	0	0
$310$	0	0
$311$	−9.08801	−0.515334	−0.257667	−	0.966234i	$-0.582954\pi$
$311$	−9.08801	−0.515334	−0.257667	+	0.966234i	$0.582954\pi$
$312$	0	0
$313$	17.5440	0.991646	0.495823	−	0.868424i	$-0.334867\pi$
$313$	17.5440	0.991646	0.495823	+	0.868424i	$0.334867\pi$
$314$	15.3160	0.864332
$315$	0	0
$316$	−1.77200	−0.0996829
$317$	−25.5440	−1.43469	−0.717347	−	0.696716i	$-0.754644\pi$
$317$	−25.5440	−1.43469	−0.717347	+	0.696716i	$0.754644\pi$
$318$	0	0
$319$	−11.3160	−0.633575
$320$	0	0
$321$	0	0
$322$	3.77200	0.210205
$323$	7.54400	0.419760
$324$	0	0
$325$	0	0
$326$	−10.2280	−0.566476
$327$	0	0
$328$	−6.77200	−0.373921
$329$	−4.54400	−0.250519
$330$	0	0
$331$	−19.8600	−1.09160	−0.545802	−	0.837914i	$-0.683775\pi$
$331$	−19.8600	−1.09160	−0.545802	+	0.837914i	$0.683775\pi$
$332$	0	0
$333$	0	0
$334$	−5.22800	−0.286063
$335$	0	0
$336$	0	0
$337$	4.00000	0.217894	0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000	0.217894	0.108947	+	0.994048i	$0.465252\pi$
$338$	20.3160	1.10505
$339$	0	0
$340$	0	0
$341$	−32.2280	−1.74524
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	1.77200	0.0955399
$345$	0	0
$346$	0.772002	0.0415031
$347$	14.3160	0.768523	0.384262	−	0.923224i	$-0.374456\pi$
$347$	14.3160	0.768523	0.384262	+	0.923224i	$0.374456\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	0	0
$352$	3.77200	0.201048
$353$	11.3160	0.602290	0.301145	−	0.953578i	$-0.402631\pi$
$353$	11.3160	0.602290	0.301145	+	0.953578i	$0.402631\pi$
$354$	0	0
$355$	0	0
$356$	13.5440	0.717831
$357$	0	0
$358$	−1.54400	−0.0816031
$359$	4.45600	0.235178	0.117589	−	0.993062i	$-0.462483\pi$
$359$	4.45600	0.235178	0.117589	+	0.993062i	$0.462483\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	4.31601	0.226844
$363$	0	0
$364$	5.77200	0.302535
$365$	0	0
$366$	0	0
$367$	−5.08801	−0.265592	−0.132796	−	0.991143i	$-0.542395\pi$
$367$	−5.08801	−0.265592	−0.132796	+	0.991143i	$0.542395\pi$
$368$	−3.77200	−0.196629
$369$	0	0
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	−23.0000	−1.19089	−0.595447	−	0.803394i	$-0.703025\pi$
$373$	−23.0000	−1.19089	−0.595447	+	0.803394i	$0.703025\pi$
$374$	14.2280	0.735712
$375$	0	0
$376$	4.54400	0.234339
$377$	17.3160	0.891820
$378$	0	0
$379$	20.7720	1.06699	0.533493	−	0.845804i	$-0.320879\pi$
$379$	20.7720	1.06699	0.533493	+	0.845804i	$0.320879\pi$
$380$	0	0
$381$	0	0
$382$	−15.0880	−0.771970
$383$	17.3160	0.884807	0.442403	−	0.896816i	$-0.354126\pi$
$383$	17.3160	0.884807	0.442403	+	0.896816i	$0.354126\pi$
$384$	0	0
$385$	0	0
$386$	−6.45600	−0.328602
$387$	0	0
$388$	17.5440	0.890662
$389$	−3.00000	−0.152106	−0.0760530	−	0.997104i	$-0.524232\pi$
$389$	−3.00000	−0.152106	−0.0760530	+	0.997104i	$0.524232\pi$
$390$	0	0
$391$	−14.2280	−0.719541
$392$	1.00000	0.0505076
$393$	0	0
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	−22.4040	−1.12443	−0.562213	−	0.826993i	$-0.690050\pi$
$397$	−22.4040	−1.12443	−0.562213	+	0.826993i	$0.690050\pi$
$398$	25.3160	1.26898
$399$	0	0
$400$	0	0
$401$	22.6320	1.13019	0.565094	−	0.825026i	$-0.308840\pi$
$401$	22.6320	1.13019	0.565094	+	0.825026i	$0.308840\pi$
$402$	0	0
$403$	49.3160	2.45661
$404$	17.3160	0.861503
$405$	0	0
$406$	3.00000	0.148888
$407$	−33.0880	−1.64011
$408$	0	0
$409$	−12.2280	−0.604636	−0.302318	−	0.953207i	$-0.597760\pi$
$409$	−12.2280	−0.604636	−0.302318	+	0.953207i	$0.597760\pi$
$410$	0	0
$411$	0	0
$412$	−8.00000	−0.394132
$413$	−6.77200	−0.333228
$414$	0	0
$415$	0	0
$416$	−5.77200	−0.282996
$417$	0	0
$418$	7.54400	0.368989
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	−17.5440	−0.855042	−0.427521	−	0.904005i	$-0.640613\pi$
$421$	−17.5440	−0.855042	−0.427521	+	0.904005i	$0.640613\pi$
$422$	10.3160	0.502175
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	−2.77200	−0.134147
$428$	−18.7720	−0.907379
$429$	0	0
$430$	0	0
$431$	−9.68399	−0.466462	−0.233231	−	0.972421i	$-0.574930\pi$
$431$	−9.68399	−0.466462	−0.233231	+	0.972421i	$0.574930\pi$
$432$	0	0
$433$	16.7720	0.806011	0.403005	−	0.915198i	$-0.367966\pi$
$433$	16.7720	0.806011	0.403005	+	0.915198i	$0.367966\pi$
$434$	8.54400	0.410125
$435$	0	0
$436$	0.455996	0.0218383
$437$	−7.54400	−0.360879
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	0	0
$442$	−21.7720	−1.03559
$443$	−9.86001	−0.468463	−0.234232	−	0.972181i	$-0.575257\pi$
$443$	−9.86001	−0.468463	−0.234232	+	0.972181i	$0.575257\pi$
$444$	0	0
$445$	0	0
$446$	2.45600	0.116295
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	15.0880	0.712047	0.356024	−	0.934477i	$-0.384132\pi$
$449$	15.0880	0.712047	0.356024	+	0.934477i	$0.384132\pi$
$450$	0	0
$451$	−25.5440	−1.20282
$452$	16.5440	0.778164
$453$	0	0
$454$	−24.0000	−1.12638
$455$	0	0
$456$	0	0
$457$	35.5440	1.66268	0.831339	−	0.555765i	$-0.187575\pi$
$457$	35.5440	1.66268	0.831339	+	0.555765i	$0.187575\pi$
$458$	−5.54400	−0.259054
$459$	0	0
$460$	0	0
$461$	−16.4560	−0.766432	−0.383216	−	0.923659i	$-0.625184\pi$
$461$	−16.4560	−0.766432	−0.383216	+	0.923659i	$0.625184\pi$
$462$	0	0
$463$	6.31601	0.293530	0.146765	−	0.989171i	$-0.453114\pi$
$463$	6.31601	0.293530	0.146765	+	0.989171i	$0.453114\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−24.7720	−1.14754
$467$	37.5440	1.73733	0.868665	−	0.495401i	$-0.164979\pi$
$467$	37.5440	1.73733	0.868665	+	0.495401i	$0.164979\pi$
$468$	0	0
$469$	9.54400	0.440701
$470$	0	0
$471$	0	0
$472$	6.77200	0.311707
$473$	6.68399	0.307330
$474$	0	0
$475$	0	0
$476$	−3.77200	−0.172889
$477$	0	0
$478$	21.8600	0.999854
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	50.6320	2.30862
$482$	−19.7720	−0.900590
$483$	0	0
$484$	3.22800	0.146727
$485$	0	0
$486$	0	0
$487$	−11.8600	−0.537428	−0.268714	−	0.963220i	$-0.586599\pi$
$487$	−11.8600	−0.537428	−0.268714	+	0.963220i	$0.586599\pi$
$488$	2.77200	0.125483
$489$	0	0
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	−11.3160	−0.509647
$494$	−11.5440	−0.519389
$495$	0	0
$496$	−8.54400	−0.383637
$497$	6.77200	0.303766
$498$	0	0
$499$	37.4040	1.67443	0.837217	−	0.546871i	$-0.184181\pi$
$499$	37.4040	1.67443	0.837217	+	0.546871i	$0.184181\pi$
$500$	0	0
$501$	0	0
$502$	−2.22800	−0.0994404
$503$	−9.77200	−0.435712	−0.217856	−	0.975981i	$-0.569906\pi$
$503$	−9.77200	−0.435712	−0.217856	+	0.975981i	$0.569906\pi$
$504$	0	0
$505$	0	0
$506$	−14.2280	−0.632512
$507$	0	0
$508$	−7.22800	−0.320691
$509$	−32.4040	−1.43628	−0.718141	−	0.695897i	$-0.755007\pi$
$509$	−32.4040	−1.43628	−0.718141	+	0.695897i	$0.755007\pi$
$510$	0	0
$511$	14.7720	0.653475
$512$	1.00000	0.0441942
$513$	0	0
$514$	−24.8600	−1.09653
$515$	0	0
$516$	0	0
$517$	17.1400	0.753816
$518$	8.77200	0.385420
$519$	0	0
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	−13.3160	−0.582268	−0.291134	−	0.956682i	$-0.594033\pi$
$523$	−13.3160	−0.582268	−0.291134	+	0.956682i	$0.594033\pi$
$524$	−3.00000	−0.131056
$525$	0	0
$526$	−24.0000	−1.04645
$527$	−32.2280	−1.40387
$528$	0	0
$529$	−8.77200	−0.381391
$530$	0	0
$531$	0	0
$532$	−2.00000	−0.0867110
$533$	39.0880	1.69309
$534$	0	0
$535$	0	0
$536$	−9.54400	−0.412238
$537$	0	0
$538$	0.683994	0.0294891
$539$	3.77200	0.162472
$540$	0	0
$541$	24.6320	1.05901	0.529506	−	0.848306i	$-0.322377\pi$
$541$	24.6320	1.05901	0.529506	+	0.848306i	$0.322377\pi$
$542$	−27.4040	−1.17710
$543$	0	0
$544$	3.77200	0.161723
$545$	0	0
$546$	0	0
$547$	4.86001	0.207799	0.103899	−	0.994588i	$-0.466868\pi$
$547$	4.86001	0.207799	0.103899	+	0.994588i	$0.466868\pi$
$548$	−12.7720	−0.545593
$549$	0	0
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	1.77200	0.0753532
$554$	22.0000	0.934690
$555$	0	0
$556$	−10.0000	−0.424094
$557$	25.5440	1.08233	0.541167	−	0.840915i	$-0.317983\pi$
$557$	25.5440	1.08233	0.541167	+	0.840915i	$0.317983\pi$
$558$	0	0
$559$	−10.2280	−0.432598
$560$	0	0
$561$	0	0
$562$	−25.5440	−1.07751
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	0	0
$566$	−4.31601	−0.181415
$567$	0	0
$568$	−6.77200	−0.284147
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	−13.8600	−0.580023	−0.290012	−	0.957023i	$-0.593659\pi$
$571$	−13.8600	−0.580023	−0.290012	+	0.957023i	$0.593659\pi$
$572$	−21.7720	−0.910333
$573$	0	0
$574$	6.77200	0.282658
$575$	0	0
$576$	0	0
$577$	40.9480	1.70469	0.852344	−	0.522981i	$-0.175180\pi$
$577$	40.9480	1.70469	0.852344	+	0.522981i	$0.175180\pi$
$578$	−2.77200	−0.115300
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−22.6320	−0.937322
$584$	−14.7720	−0.611270
$585$	0	0
$586$	3.68399	0.152184
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	0	0
$589$	−17.0880	−0.704099
$590$	0	0
$591$	0	0
$592$	−8.77200	−0.360527
$593$	17.3160	0.711083	0.355542	−	0.934660i	$-0.384296\pi$
$593$	17.3160	0.711083	0.355542	+	0.934660i	$0.384296\pi$
$594$	0	0
$595$	0	0
$596$	−11.3160	−0.463522
$597$	0	0
$598$	21.7720	0.890323
$599$	39.8600	1.62864	0.814318	−	0.580419i	$-0.197111\pi$
$599$	39.8600	1.62864	0.814318	+	0.580419i	$0.197111\pi$
$600$	0	0
$601$	2.68399	0.109482	0.0547412	−	0.998501i	$-0.482567\pi$
$601$	2.68399	0.109482	0.0547412	+	0.998501i	$0.482567\pi$
$602$	−1.77200	−0.0722214
$603$	0	0
$604$	1.31601	0.0535475
$605$	0	0
$606$	0	0
$607$	41.7200	1.69336	0.846682	−	0.532100i	$-0.178597\pi$
$607$	41.7200	1.69336	0.846682	+	0.532100i	$0.178597\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	0	0
$611$	−26.2280	−1.06107
$612$	0	0
$613$	39.3160	1.58796	0.793979	−	0.607945i	$-0.208006\pi$
$613$	39.3160	1.58796	0.793979	+	0.607945i	$0.208006\pi$
$614$	33.3160	1.34452
$615$	0	0
$616$	−3.77200	−0.151978
$617$	39.1760	1.57717	0.788583	−	0.614928i	$-0.210815\pi$
$617$	39.1760	1.57717	0.788583	+	0.614928i	$0.210815\pi$
$618$	0	0
$619$	0.455996	0.0183280	0.00916402	−	0.999958i	$-0.497083\pi$
$619$	0.455996	0.0183280	0.00916402	+	0.999958i	$0.497083\pi$
$620$	0	0
$621$	0	0
$622$	−9.08801	−0.364396
$623$	−13.5440	−0.542629
$624$	0	0
$625$	0	0
$626$	17.5440	0.701199
$627$	0	0
$628$	15.3160	0.611175
$629$	−33.0880	−1.31931
$630$	0	0
$631$	21.5440	0.857653	0.428827	−	0.903387i	$-0.358927\pi$
$631$	21.5440	0.857653	0.428827	+	0.903387i	$0.358927\pi$
$632$	−1.77200	−0.0704864
$633$	0	0
$634$	−25.5440	−1.01448
$635$	0	0
$636$	0	0
$637$	−5.77200	−0.228695
$638$	−11.3160	−0.448005
$639$	0	0
$640$	0	0
$641$	−34.6320	−1.36788	−0.683941	−	0.729537i	$-0.739735\pi$
$641$	−34.6320	−1.36788	−0.683941	+	0.729537i	$0.739735\pi$
$642$	0	0
$643$	43.1760	1.70270	0.851348	−	0.524602i	$-0.175786\pi$
$643$	43.1760	1.70270	0.851348	+	0.524602i	$0.175786\pi$
$644$	3.77200	0.148638
$645$	0	0
$646$	7.54400	0.296815
$647$	−23.4040	−0.920107	−0.460053	−	0.887891i	$-0.652170\pi$
$647$	−23.4040	−0.920107	−0.460053	+	0.887891i	$0.652170\pi$
$648$	0	0
$649$	25.5440	1.00269
$650$	0	0
$651$	0	0
$652$	−10.2280	−0.400559
$653$	33.0880	1.29483	0.647417	−	0.762136i	$-0.275849\pi$
$653$	33.0880	1.29483	0.647417	+	0.762136i	$0.275849\pi$
$654$	0	0
$655$	0	0
$656$	−6.77200	−0.264402
$657$	0	0
$658$	−4.54400	−0.177144
$659$	−42.1760	−1.64294	−0.821472	−	0.570249i	$-0.806847\pi$
$659$	−42.1760	−1.64294	−0.821472	+	0.570249i	$0.806847\pi$
$660$	0	0
$661$	−1.68399	−0.0654998	−0.0327499	−	0.999464i	$-0.510426\pi$
$661$	−1.68399	−0.0654998	−0.0327499	+	0.999464i	$0.510426\pi$
$662$	−19.8600	−0.771881
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11.3160	0.438157
$668$	−5.22800	−0.202277
$669$	0	0
$670$	0	0
$671$	10.4560	0.403649
$672$	0	0
$673$	−48.6320	−1.87463	−0.937313	−	0.348488i	$-0.886695\pi$
$673$	−48.6320	−1.87463	−0.937313	+	0.348488i	$0.886695\pi$
$674$	4.00000	0.154074
$675$	0	0
$676$	20.3160	0.781385
$677$	15.6840	0.602785	0.301392	−	0.953500i	$-0.402549\pi$
$677$	15.6840	0.602785	0.301392	+	0.953500i	$0.402549\pi$
$678$	0	0
$679$	−17.5440	−0.673277
$680$	0	0
$681$	0	0
$682$	−32.2280	−1.23407
$683$	−23.2280	−0.888795	−0.444397	−	0.895830i	$-0.646582\pi$
$683$	−23.2280	−0.888795	−0.444397	+	0.895830i	$0.646582\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	1.77200	0.0675569
$689$	34.6320	1.31937
$690$	0	0
$691$	41.0880	1.56306	0.781531	−	0.623867i	$-0.214439\pi$
$691$	41.0880	1.56306	0.781531	+	0.623867i	$0.214439\pi$
$692$	0.772002	0.0293471
$693$	0	0
$694$	14.3160	0.543428
$695$	0	0
$696$	0	0
$697$	−25.5440	−0.967548
$698$	−22.0000	−0.832712
$699$	0	0
$700$	0	0
$701$	−15.0000	−0.566542	−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000	−0.566542	−0.283271	+	0.959040i	$0.591420\pi$
$702$	0	0
$703$	−17.5440	−0.661685
$704$	3.77200	0.142163
$705$	0	0
$706$	11.3160	0.425883
$707$	−17.3160	−0.651235
$708$	0	0
$709$	18.4560	0.693129	0.346565	−	0.938026i	$-0.387348\pi$
$709$	18.4560	0.693129	0.346565	+	0.938026i	$0.387348\pi$
$710$	0	0
$711$	0	0
$712$	13.5440	0.507583
$713$	32.2280	1.20695
$714$	0	0
$715$	0	0
$716$	−1.54400	−0.0577021
$717$	0	0
$718$	4.45600	0.166296
$719$	−40.6320	−1.51532	−0.757659	−	0.652650i	$-0.773657\pi$
$719$	−40.6320	−1.51532	−0.757659	+	0.652650i	$0.773657\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	−15.0000	−0.558242
$723$	0	0
$724$	4.31601	0.160403
$725$	0	0
$726$	0	0
$727$	−3.54400	−0.131440	−0.0657199	−	0.997838i	$-0.520934\pi$
$727$	−3.54400	−0.131440	−0.0657199	+	0.997838i	$0.520934\pi$
$728$	5.77200	0.213925
$729$	0	0
$730$	0	0
$731$	6.68399	0.247216
$732$	0	0
$733$	−39.5440	−1.46059	−0.730296	−	0.683131i	$-0.760618\pi$
$733$	−39.5440	−1.46059	−0.730296	+	0.683131i	$0.760618\pi$
$734$	−5.08801	−0.187802
$735$	0	0
$736$	−3.77200	−0.139038
$737$	−36.0000	−1.32608
$738$	0	0
$739$	7.22800	0.265886	0.132943	−	0.991124i	$-0.457557\pi$
$739$	7.22800	0.265886	0.132943	+	0.991124i	$0.457557\pi$
$740$	0	0
$741$	0	0
$742$	6.00000	0.220267
$743$	12.6840	0.465331	0.232665	−	0.972557i	$-0.425255\pi$
$743$	12.6840	0.465331	0.232665	+	0.972557i	$0.425255\pi$
$744$	0	0
$745$	0	0
$746$	−23.0000	−0.842090
$747$	0	0
$748$	14.2280	0.520227
$749$	18.7720	0.685914
$750$	0	0
$751$	35.7720	1.30534	0.652669	−	0.757643i	$-0.273649\pi$
$751$	35.7720	1.30534	0.652669	+	0.757643i	$0.273649\pi$
$752$	4.54400	0.165703
$753$	0	0
$754$	17.3160	0.630612
$755$	0	0
$756$	0	0
$757$	−51.6320	−1.87660	−0.938299	−	0.345826i	$-0.887599\pi$
$757$	−51.6320	−1.87660	−0.938299	+	0.345826i	$0.887599\pi$
$758$	20.7720	0.754473
$759$	0	0
$760$	0	0
$761$	−12.7720	−0.462985	−0.231492	−	0.972837i	$-0.574361\pi$
$761$	−12.7720	−0.462985	−0.231492	+	0.972837i	$0.574361\pi$
$762$	0	0
$763$	−0.455996	−0.0165082
$764$	−15.0880	−0.545865
$765$	0	0
$766$	17.3160	0.625653
$767$	−39.0880	−1.41139
$768$	0	0
$769$	−36.2280	−1.30642	−0.653208	−	0.757179i	$-0.726577\pi$
$769$	−36.2280	−1.30642	−0.653208	+	0.757179i	$0.726577\pi$
$770$	0	0
$771$	0	0
$772$	−6.45600	−0.232356
$773$	15.8600	0.570445	0.285222	−	0.958461i	$-0.407933\pi$
$773$	15.8600	0.570445	0.285222	+	0.958461i	$0.407933\pi$
$774$	0	0
$775$	0	0
$776$	17.5440	0.629793
$777$	0	0
$778$	−3.00000	−0.107555
$779$	−13.5440	−0.485264
$780$	0	0
$781$	−25.5440	−0.914036
$782$	−14.2280	−0.508792
$783$	0	0
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	−27.6320	−0.984975	−0.492487	−	0.870320i	$-0.663912\pi$
$787$	−27.6320	−0.984975	−0.492487	+	0.870320i	$0.663912\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	0	0
$791$	−16.5440	−0.588237
$792$	0	0
$793$	−16.0000	−0.568177
$794$	−22.4040	−0.795089
$795$	0	0
$796$	25.3160	0.897302
$797$	−38.3160	−1.35722	−0.678611	−	0.734498i	$-0.737418\pi$
$797$	−38.3160	−1.35722	−0.678611	+	0.734498i	$0.737418\pi$
$798$	0	0
$799$	17.1400	0.606369
$800$	0	0
$801$	0	0
$802$	22.6320	0.799164
$803$	−55.7200	−1.96632
$804$	0	0
$805$	0	0
$806$	49.3160	1.73708
$807$	0	0
$808$	17.3160	0.609175
$809$	43.7200	1.53711	0.768557	−	0.639781i	$-0.220975\pi$
$809$	43.7200	1.53711	0.768557	+	0.639781i	$0.220975\pi$
$810$	0	0
$811$	4.91199	0.172483	0.0862417	−	0.996274i	$-0.472514\pi$
$811$	4.91199	0.172483	0.0862417	+	0.996274i	$0.472514\pi$
$812$	3.00000	0.105279
$813$	0	0
$814$	−33.0880	−1.15973
$815$	0	0
$816$	0	0
$817$	3.54400	0.123989
$818$	−12.2280	−0.427542
$819$	0	0
$820$	0	0
$821$	9.08801	0.317174	0.158587	−	0.987345i	$-0.449306\pi$
$821$	9.08801	0.317174	0.158587	+	0.987345i	$0.449306\pi$
$822$	0	0
$823$	−25.4040	−0.885528	−0.442764	−	0.896638i	$-0.646002\pi$
$823$	−25.4040	−0.885528	−0.442764	+	0.896638i	$0.646002\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	−6.77200	−0.235628
$827$	−14.3160	−0.497816	−0.248908	−	0.968527i	$-0.580072\pi$
$827$	−14.3160	−0.497816	−0.248908	+	0.968527i	$0.580072\pi$
$828$	0	0
$829$	−20.6320	−0.716579	−0.358290	−	0.933610i	$-0.616640\pi$
$829$	−20.6320	−0.716579	−0.358290	+	0.933610i	$0.616640\pi$
$830$	0	0
$831$	0	0
$832$	−5.77200	−0.200108
$833$	3.77200	0.130692
$834$	0	0
$835$	0	0
$836$	7.54400	0.260915
$837$	0	0
$838$	−3.00000	−0.103633
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−17.5440	−0.604606
$843$	0	0
$844$	10.3160	0.355092
$845$	0	0
$846$	0	0
$847$	−3.22800	−0.110915
$848$	−6.00000	−0.206041
$849$	0	0
$850$	0	0
$851$	33.0880	1.13424
$852$	0	0
$853$	10.8600	0.371840	0.185920	−	0.982565i	$-0.440474\pi$
$853$	10.8600	0.371840	0.185920	+	0.982565i	$0.440474\pi$
$854$	−2.77200	−0.0948560
$855$	0	0
$856$	−18.7720	−0.641614
$857$	49.7200	1.69840	0.849202	−	0.528069i	$-0.177084\pi$
$857$	49.7200	1.69840	0.849202	+	0.528069i	$0.177084\pi$
$858$	0	0
$859$	38.0000	1.29654	0.648272	−	0.761409i	$-0.275492\pi$
$859$	38.0000	1.29654	0.648272	+	0.761409i	$0.275492\pi$
$860$	0	0
$861$	0	0
$862$	−9.68399	−0.329838
$863$	−6.68399	−0.227526	−0.113763	−	0.993508i	$-0.536290\pi$
$863$	−6.68399	−0.227526	−0.113763	+	0.993508i	$0.536290\pi$
$864$	0	0
$865$	0	0
$866$	16.7720	0.569936
$867$	0	0
$868$	8.54400	0.290002
$869$	−6.68399	−0.226739
$870$	0	0
$871$	55.0880	1.86659
$872$	0.455996	0.0154420
$873$	0	0
$874$	−7.54400	−0.255180
$875$	0	0
$876$	0	0
$877$	17.6320	0.595391	0.297695	−	0.954661i	$-0.403782\pi$
$877$	17.6320	0.595391	0.297695	+	0.954661i	$0.403782\pi$
$878$	8.00000	0.269987
$879$	0	0
$880$	0	0
$881$	8.31601	0.280173	0.140087	−	0.990139i	$-0.455262\pi$
$881$	8.31601	0.280173	0.140087	+	0.990139i	$0.455262\pi$
$882$	0	0
$883$	0.911993	0.0306910	0.0153455	−	0.999882i	$-0.495115\pi$
$883$	0.911993	0.0306910	0.0153455	+	0.999882i	$0.495115\pi$
$884$	−21.7720	−0.732272
$885$	0	0
$886$	−9.86001	−0.331253
$887$	16.5440	0.555493	0.277747	−	0.960654i	$-0.410412\pi$
$887$	16.5440	0.555493	0.277747	+	0.960654i	$0.410412\pi$
$888$	0	0
$889$	7.22800	0.242419
$890$	0	0
$891$	0	0
$892$	2.45600	0.0822328
$893$	9.08801	0.304119
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	15.0880	0.503493
$899$	25.6320	0.854875
$900$	0	0
$901$	−22.6320	−0.753982
$902$	−25.5440	−0.850522
$903$	0	0
$904$	16.5440	0.550245
$905$	0	0
$906$	0	0
$907$	37.7720	1.25420	0.627099	−	0.778939i	$-0.284242\pi$
$907$	37.7720	1.25420	0.627099	+	0.778939i	$0.284242\pi$
$908$	−24.0000	−0.796468
$909$	0	0
$910$	0	0
$911$	32.3160	1.07068	0.535339	−	0.844638i	$-0.320184\pi$
$911$	32.3160	1.07068	0.535339	+	0.844638i	$0.320184\pi$
$912$	0	0
$913$	0	0
$914$	35.5440	1.17569
$915$	0	0
$916$	−5.54400	−0.183179
$917$	3.00000	0.0990687
$918$	0	0
$919$	−45.4920	−1.50064	−0.750322	−	0.661073i	$-0.770101\pi$
$919$	−45.4920	−1.50064	−0.750322	+	0.661073i	$0.770101\pi$
$920$	0	0
$921$	0	0
$922$	−16.4560	−0.541949
$923$	39.0880	1.28660
$924$	0	0
$925$	0	0
$926$	6.31601	0.207557
$927$	0	0
$928$	−3.00000	−0.0984798
$929$	21.0880	0.691875	0.345938	−	0.938258i	$-0.387561\pi$
$929$	21.0880	0.691875	0.345938	+	0.938258i	$0.387561\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	−24.7720	−0.811434
$933$	0	0
$934$	37.5440	1.22848
$935$	0	0
$936$	0	0
$937$	6.13999	0.200585	0.100292	−	0.994958i	$-0.468022\pi$
$937$	6.13999	0.200585	0.100292	+	0.994958i	$0.468022\pi$
$938$	9.54400	0.311623
$939$	0	0
$940$	0	0
$941$	23.3160	0.760080	0.380040	−	0.924970i	$-0.375910\pi$
$941$	23.3160	0.760080	0.380040	+	0.924970i	$0.375910\pi$
$942$	0	0
$943$	25.5440	0.831827
$944$	6.77200	0.220410
$945$	0	0
$946$	6.68399	0.217315
$947$	57.2640	1.86083	0.930415	−	0.366507i	$-0.119446\pi$
$947$	57.2640	1.86083	0.930415	+	0.366507i	$0.119446\pi$
$948$	0	0
$949$	85.2640	2.76779
$950$	0	0
$951$	0	0
$952$	−3.77200	−0.122251
$953$	−59.3160	−1.92143	−0.960717	−	0.277530i	$-0.910484\pi$
$953$	−59.3160	−1.92143	−0.960717	+	0.277530i	$0.910484\pi$
$954$	0	0
$955$	0	0
$956$	21.8600	0.707003
$957$	0	0
$958$	36.0000	1.16311
$959$	12.7720	0.412429
$960$	0	0
$961$	42.0000	1.35484
$962$	50.6320	1.63244
$963$	0	0
$964$	−19.7720	−0.636813
$965$	0	0
$966$	0	0
$967$	−20.7720	−0.667983	−0.333991	−	0.942576i	$-0.608396\pi$
$967$	−20.7720	−0.667983	−0.333991	+	0.942576i	$0.608396\pi$
$968$	3.22800	0.103752
$969$	0	0
$970$	0	0
$971$	−45.0000	−1.44412	−0.722059	−	0.691831i	$-0.756804\pi$
$971$	−45.0000	−1.44412	−0.722059	+	0.691831i	$0.756804\pi$
$972$	0	0
$973$	10.0000	0.320585
$974$	−11.8600	−0.380019
$975$	0	0
$976$	2.77200	0.0887296
$977$	35.3160	1.12986	0.564930	−	0.825139i	$-0.308903\pi$
$977$	35.3160	1.12986	0.564930	+	0.825139i	$0.308903\pi$
$978$	0	0
$979$	51.0880	1.63278
$980$	0	0
$981$	0	0
$982$	−12.0000	−0.382935
$983$	−36.8600	−1.17565	−0.587826	−	0.808987i	$-0.700016\pi$
$983$	−36.8600	−1.17565	−0.587826	+	0.808987i	$0.700016\pi$
$984$	0	0
$985$	0	0
$986$	−11.3160	−0.360375
$987$	0	0
$988$	−11.5440	−0.367264
$989$	−6.68399	−0.212539
$990$	0	0
$991$	10.4040	0.330494	0.165247	−	0.986252i	$-0.447158\pi$
$991$	10.4040	0.330494	0.165247	+	0.986252i	$0.447158\pi$
$992$	−8.54400	−0.271272
$993$	0	0
$994$	6.77200	0.214795
$995$	0	0
$996$	0	0
$997$	3.31601	0.105019	0.0525095	−	0.998620i	$-0.483278\pi$
$997$	3.31601	0.105019	0.0525095	+	0.998620i	$0.483278\pi$
$998$	37.4040	1.18400
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9450.2.a.eo.1.2		2
3.2	odd	2		9450.2.a.ed.1.1		2
5.4	even	2		1890.2.a.z.1.2	✓	2
15.14	odd	2		1890.2.a.bb.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1890.2.a.z.1.2	✓	2	5.4	even	2
1890.2.a.bb.1.1	yes	2	15.14	odd	2
9450.2.a.ed.1.1		2	3.2	odd	2
9450.2.a.eo.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} +3.77200 q^{11} -5.77200 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.77200 q^{17} +2.00000 q^{19} +3.77200 q^{22} -3.77200 q^{23} -5.77200 q^{26} -1.00000 q^{28} -3.00000 q^{29} -8.54400 q^{31} +1.00000 q^{32} +3.77200 q^{34} -8.77200 q^{37} +2.00000 q^{38} -6.77200 q^{41} +1.77200 q^{43} +3.77200 q^{44} -3.77200 q^{46} +4.54400 q^{47} +1.00000 q^{49} -5.77200 q^{52} -6.00000 q^{53} -1.00000 q^{56} -3.00000 q^{58} +6.77200 q^{59} +2.77200 q^{61} -8.54400 q^{62} +1.00000 q^{64} -9.54400 q^{67} +3.77200 q^{68} -6.77200 q^{71} -14.7720 q^{73} -8.77200 q^{74} +2.00000 q^{76} -3.77200 q^{77} -1.77200 q^{79} -6.77200 q^{82} +1.77200 q^{86} +3.77200 q^{88} +13.5440 q^{89} +5.77200 q^{91} -3.77200 q^{92} +4.54400 q^{94} +17.5440 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} - q^{11} - 3 q^{13} - 2 q^{14} + 2 q^{16} - q^{17} + 4 q^{19} - q^{22} + q^{23} - 3 q^{26} - 2 q^{28} - 6 q^{29} + 2 q^{32} - q^{34} - 9 q^{37} + 4 q^{38} - 5 q^{41} - 5 q^{43} - q^{44} + q^{46} - 8 q^{47} + 2 q^{49} - 3 q^{52} - 12 q^{53} - 2 q^{56} - 6 q^{58} + 5 q^{59} - 3 q^{61} + 2 q^{64} - 2 q^{67} - q^{68} - 5 q^{71} - 21 q^{73} - 9 q^{74} + 4 q^{76} + q^{77} + 5 q^{79} - 5 q^{82} - 5 q^{86} - q^{88} + 10 q^{89} + 3 q^{91} + q^{92} - 8 q^{94} + 18 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 - q^11 - 3 * q^13 - 2 * q^14 + 2 * q^16 - q^17 + 4 * q^19 - q^22 + q^23 - 3 * q^26 - 2 * q^28 - 6 * q^29 + 2 * q^32 - q^34 - 9 * q^37 + 4 * q^38 - 5 * q^41 - 5 * q^43 - q^44 + q^46 - 8 * q^47 + 2 * q^49 - 3 * q^52 - 12 * q^53 - 2 * q^56 - 6 * q^58 + 5 * q^59 - 3 * q^61 + 2 * q^64 - 2 * q^67 - q^68 - 5 * q^71 - 21 * q^73 - 9 * q^74 + 4 * q^76 + q^77 + 5 * q^79 - 5 * q^82 - 5 * q^86 - q^88 + 10 * q^89 + 3 * q^91 + q^92 - 8 * q^94 + 18 * q^97 + 2 * q^98

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